erdsze2lem2 - Mathbox for Mario Carneiro

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Theorem erdsze2lem2 36177

Description: Lemma for erdsze2 36178. (Contributed by Mario Carneiro, 22-Jan-2015.)

**Hypotheses**
Ref	Expression
erdsze2.r	⊢ (𝜑 → 𝑅 ∈ ℕ)
erdsze2.s	⊢ (𝜑 → 𝑆 ∈ ℕ)
erdsze2.f	⊢ (𝜑 → 𝐹:𝐴⟶1-1ℝ)
erdsze2.a	⊢ (𝜑 → 𝐴 ⊆ ℝ)
erdsze2lem.n	⊢ 𝑁 = ((𝑅 − 1) · (𝑆 − 1))
erdsze2lem.l	⊢ (𝜑 → 𝑁 < (♯‘𝐴))
erdsze2lem.g	⊢ (𝜑 → 𝐺:(1...(𝑁 + 1))⟶1-1𝐴)
erdsze2lem.i	⊢ (𝜑 → 𝐺 Isom < , < ((1...(𝑁 + 1)), ran 𝐺))

**Assertion**
Ref	Expression
erdsze2lem2	⊢ (𝜑 → ∃𝑠 ∈ 𝒫 𝐴((𝑅 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , < (𝑠, (𝐹 “ 𝑠))) ∨ (𝑆 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , ^◡ < (𝑠, (𝐹 “ 𝑠)))))

Distinct variable groups: 𝐴,𝑠 𝐹,𝑠 𝐺,𝑠 𝑅,𝑠 𝑆,𝑠 𝑁,𝑠 𝜑,𝑠

Proof of Theorem erdsze2lem2 Dummy variables 𝑡 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		erdsze2lem.n	. . . . 5 ⊢ 𝑁 = ((𝑅 − 1) · (𝑆 − 1))
2		erdsze2.r	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ ℕ)
3		nnm1nn0 12687	. . . . . . 7 ⊢ (𝑅 ∈ ℕ → (𝑅 − 1) ∈ ℕ₀)
4	2, 3	syl 18	. . . . . 6 ⊢ (𝜑 → (𝑅 − 1) ∈ ℕ₀)
5		erdsze2.s	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ ℕ)
6		nnm1nn0 12687	. . . . . . 7 ⊢ (𝑆 ∈ ℕ → (𝑆 − 1) ∈ ℕ₀)
7	5, 6	syl 18	. . . . . 6 ⊢ (𝜑 → (𝑆 − 1) ∈ ℕ₀)
8	4, 7	nn0mulcld 12712	. . . . 5 ⊢ (𝜑 → ((𝑅 − 1) · (𝑆 − 1)) ∈ ℕ₀)
9	1, 8	eqeltrid 2868	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
10		nn0p1nn 12685	. . . 4 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ ℕ)
11	9, 10	syl 18	. . 3 ⊢ (𝜑 → (𝑁 + 1) ∈ ℕ)
12		erdsze2.f	. . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶1-1ℝ)
13		erdsze2lem.g	. . . 4 ⊢ (𝜑 → 𝐺:(1...(𝑁 + 1))⟶1-1𝐴)
14		f1co 6821	. . . 4 ⊢ ((𝐹:𝐴⟶1-1ℝ ∧ 𝐺:(1...(𝑁 + 1))⟶1-1𝐴) → (𝐹 ∘ 𝐺):(1...(𝑁 + 1))⟶1-1ℝ)
15	12, 13, 14	syl2anc 596	. . 3 ⊢ (𝜑 → (𝐹 ∘ 𝐺):(1...(𝑁 + 1))⟶1-1ℝ)
16	9	nn0red 12708	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℝ)
17	16	ltp1d 12287	. . . 4 ⊢ (𝜑 → 𝑁 < (𝑁 + 1))
18	1, 17	eqbrtrrid 5147	. . 3 ⊢ (𝜑 → ((𝑅 − 1) · (𝑆 − 1)) < (𝑁 + 1))
19	11, 15, 2, 5, 18	erdsze 36175	. 2 ⊢ (𝜑 → ∃𝑡 ∈ 𝒫 (1...(𝑁 + 1))((𝑅 ≤ (♯‘𝑡) ∧ ((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡))) ∨ (𝑆 ≤ (♯‘𝑡) ∧ ((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , ^◡ < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡)))))
20		velpw 4565	. . . 4 ⊢ (𝑡 ∈ 𝒫 (1...(𝑁 + 1)) ↔ 𝑡 ⊆ (1...(𝑁 + 1)))
21		imassrn 6216	. . . . . . . 8 ⊢ (𝐺 “ 𝑡) ⊆ ran 𝐺
22		f1f 6808	. . . . . . . . . 10 ⊢ (𝐺:(1...(𝑁 + 1))⟶1-1𝐴 → 𝐺:(1...(𝑁 + 1))⟶𝐴)
23	13, 22	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝐺:(1...(𝑁 + 1))⟶𝐴)
24	23	frnd 6748	. . . . . . . 8 ⊢ (𝜑 → ran 𝐺 ⊆ 𝐴)
25	21, 24	sstrid 3924	. . . . . . 7 ⊢ (𝜑 → (𝐺 “ 𝑡) ⊆ 𝐴)
26		erdsze2.a	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
27		reex 11330	. . . . . . . . 9 ⊢ ℝ ∈ V
28		ssexg 5288	. . . . . . . . 9 ⊢ ((𝐴 ⊆ ℝ ∧ ℝ ∈ V) → 𝐴 ∈ V)
29	26, 27, 28	sylancl 598	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ V)
30		elpw2g 5302	. . . . . . . 8 ⊢ (𝐴 ∈ V → ((𝐺 “ 𝑡) ∈ 𝒫 𝐴 ↔ (𝐺 “ 𝑡) ⊆ 𝐴))
31	29, 30	syl 18	. . . . . . 7 ⊢ (𝜑 → ((𝐺 “ 𝑡) ∈ 𝒫 𝐴 ↔ (𝐺 “ 𝑡) ⊆ 𝐴))
32	25, 31	mpbird 260	. . . . . 6 ⊢ (𝜑 → (𝐺 “ 𝑡) ∈ 𝒫 𝐴)
33	32	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (𝐺 “ 𝑡) ∈ 𝒫 𝐴)
34		vex 3458	. . . . . . . . . . . 12 ⊢ 𝑡 ∈ V
35	34	f1imaen 9082	. . . . . . . . . . 11 ⊢ ((𝐺:(1...(𝑁 + 1))⟶1-1𝐴 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (𝐺 “ 𝑡) ≈ 𝑡)
36	13, 35	sylan 592	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (𝐺 “ 𝑡) ≈ 𝑡)
37		fzfid 14156	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (1...(𝑁 + 1)) ∈ Fin)
38		simpr 490	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → 𝑡 ⊆ (1...(𝑁 + 1)))
39		ssfi 9226	. . . . . . . . . . . . 13 ⊢ (((1...(𝑁 + 1)) ∈ Fin ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → 𝑡 ∈ Fin)
40	37, 38, 39	syl2anc 596	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → 𝑡 ∈ Fin)
41		enfii 9239	. . . . . . . . . . . 12 ⊢ ((𝑡 ∈ Fin ∧ (𝐺 “ 𝑡) ≈ 𝑡) → (𝐺 “ 𝑡) ∈ Fin)
42	40, 36, 41	syl2anc 596	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (𝐺 “ 𝑡) ∈ Fin)
43		hashen 14532	. . . . . . . . . . 11 ⊢ (((𝐺 “ 𝑡) ∈ Fin ∧ 𝑡 ∈ Fin) → ((♯‘(𝐺 “ 𝑡)) = (♯‘𝑡) ↔ (𝐺 “ 𝑡) ≈ 𝑡))
44	42, 40, 43	syl2anc 596	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → ((♯‘(𝐺 “ 𝑡)) = (♯‘𝑡) ↔ (𝐺 “ 𝑡) ≈ 𝑡))
45	36, 44	mpbird 260	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (♯‘(𝐺 “ 𝑡)) = (♯‘𝑡))
46	45	breq2d 5120	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (𝑅 ≤ (♯‘(𝐺 “ 𝑡)) ↔ 𝑅 ≤ (♯‘𝑡)))
47	46	biimprd 251	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (𝑅 ≤ (♯‘𝑡) → 𝑅 ≤ (♯‘(𝐺 “ 𝑡))))
48		erdsze2lem.i	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐺 Isom < , < ((1...(𝑁 + 1)), ran 𝐺))
49	48	ad2antrr 739	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) ∧ (𝑥 ∈ 𝑡 ∧ 𝑦 ∈ 𝑡)) → 𝐺 Isom < , < ((1...(𝑁 + 1)), ran 𝐺))
50	38	adantr 486	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) ∧ (𝑥 ∈ 𝑡 ∧ 𝑦 ∈ 𝑡)) → 𝑡 ⊆ (1...(𝑁 + 1)))
51		simprl 783	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) ∧ (𝑥 ∈ 𝑡 ∧ 𝑦 ∈ 𝑡)) → 𝑥 ∈ 𝑡)
52	50, 51	sseldd 3914	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) ∧ (𝑥 ∈ 𝑡 ∧ 𝑦 ∈ 𝑡)) → 𝑥 ∈ (1...(𝑁 + 1)))
53		simprr 785	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) ∧ (𝑥 ∈ 𝑡 ∧ 𝑦 ∈ 𝑡)) → 𝑦 ∈ 𝑡)
54	50, 53	sseldd 3914	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) ∧ (𝑥 ∈ 𝑡 ∧ 𝑦 ∈ 𝑡)) → 𝑦 ∈ (1...(𝑁 + 1)))
55		isorel 7365	. . . . . . . . . . . . . 14 ⊢ ((𝐺 Isom < , < ((1...(𝑁 + 1)), ran 𝐺) ∧ (𝑥 ∈ (1...(𝑁 + 1)) ∧ 𝑦 ∈ (1...(𝑁 + 1)))) → (𝑥 < 𝑦 ↔ (𝐺‘𝑥) < (𝐺‘𝑦)))
56	49, 52, 54, 55	syl12anc 850	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) ∧ (𝑥 ∈ 𝑡 ∧ 𝑦 ∈ 𝑡)) → (𝑥 < 𝑦 ↔ (𝐺‘𝑥) < (𝐺‘𝑦)))
57	56	biimpd 232	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) ∧ (𝑥 ∈ 𝑡 ∧ 𝑦 ∈ 𝑡)) → (𝑥 < 𝑦 → (𝐺‘𝑥) < (𝐺‘𝑦)))
58	57	ralrimivva 3209	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥 < 𝑦 → (𝐺‘𝑥) < (𝐺‘𝑦)))
59		elfznn 13727	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 ∈ (1...(𝑁 + 1)) → 𝑡 ∈ ℕ)
60	59	nnred 12390	. . . . . . . . . . . . . . 15 ⊢ (𝑡 ∈ (1...(𝑁 + 1)) → 𝑡 ∈ ℝ)
61	60	ssriv 3917	. . . . . . . . . . . . . 14 ⊢ (1...(𝑁 + 1)) ⊆ ℝ
62	61	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (1...(𝑁 + 1)) ⊆ ℝ)
63		ltso 11429	. . . . . . . . . . . . 13 ⊢ < Or ℝ
64		soss 5586	. . . . . . . . . . . . 13 ⊢ ((1...(𝑁 + 1)) ⊆ ℝ → ( < Or ℝ → < Or (1...(𝑁 + 1))))
65	62, 63, 64	mpisyl 22	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → < Or (1...(𝑁 + 1)))
66	26	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → 𝐴 ⊆ ℝ)
67		soss 5586	. . . . . . . . . . . . 13 ⊢ (𝐴 ⊆ ℝ → ( < Or ℝ → < Or 𝐴))
68	66, 63, 67	mpisyl 22	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → < Or 𝐴)
69	23	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → 𝐺:(1...(𝑁 + 1))⟶𝐴)
70		soisores 7366	. . . . . . . . . . . 12 ⊢ ((( < Or (1...(𝑁 + 1)) ∧ < Or 𝐴) ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ 𝑡 ⊆ (1...(𝑁 + 1)))) → ((𝐺 ↾ 𝑡) Isom < , < (𝑡, (𝐺 “ 𝑡)) ↔ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥 < 𝑦 → (𝐺‘𝑥) < (𝐺‘𝑦))))
71	65, 68, 69, 38, 70	syl22anc 852	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → ((𝐺 ↾ 𝑡) Isom < , < (𝑡, (𝐺 “ 𝑡)) ↔ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥 < 𝑦 → (𝐺‘𝑥) < (𝐺‘𝑦))))
72	58, 71	mpbird 260	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (𝐺 ↾ 𝑡) Isom < , < (𝑡, (𝐺 “ 𝑡)))
73		isocnv 7369	. . . . . . . . . 10 ⊢ ((𝐺 ↾ 𝑡) Isom < , < (𝑡, (𝐺 “ 𝑡)) → ^◡(𝐺 ↾ 𝑡) Isom < , < ((𝐺 “ 𝑡), 𝑡))
74	72, 73	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → ^◡(𝐺 ↾ 𝑡) Isom < , < ((𝐺 “ 𝑡), 𝑡))
75		isotr 7375	. . . . . . . . . 10 ⊢ ((^◡(𝐺 ↾ 𝑡) Isom < , < ((𝐺 “ 𝑡), 𝑡) ∧ ((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡))) → (((𝐹 ∘ 𝐺) ↾ 𝑡) ∘ ^◡(𝐺 ↾ 𝑡)) Isom < , < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡)))
76	75	ex 418	. . . . . . . . 9 ⊢ (^◡(𝐺 ↾ 𝑡) Isom < , < ((𝐺 “ 𝑡), 𝑡) → (((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡)) → (((𝐹 ∘ 𝐺) ↾ 𝑡) ∘ ^◡(𝐺 ↾ 𝑡)) Isom < , < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡))))
77	74, 76	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡)) → (((𝐹 ∘ 𝐺) ↾ 𝑡) ∘ ^◡(𝐺 ↾ 𝑡)) Isom < , < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡))))
78		resco 6270	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∘ 𝐺) ↾ 𝑡) = (𝐹 ∘ (𝐺 ↾ 𝑡))
79	78	coeq1i 5848	. . . . . . . . . . . 12 ⊢ (((𝐹 ∘ 𝐺) ↾ 𝑡) ∘ ^◡(𝐺 ↾ 𝑡)) = ((𝐹 ∘ (𝐺 ↾ 𝑡)) ∘ ^◡(𝐺 ↾ 𝑡))
80		coass 6290	. . . . . . . . . . . 12 ⊢ ((𝐹 ∘ (𝐺 ↾ 𝑡)) ∘ ^◡(𝐺 ↾ 𝑡)) = (𝐹 ∘ ((𝐺 ↾ 𝑡) ∘ ^◡(𝐺 ↾ 𝑡)))
81	79, 80	eqtri 2786	. . . . . . . . . . 11 ⊢ (((𝐹 ∘ 𝐺) ↾ 𝑡) ∘ ^◡(𝐺 ↾ 𝑡)) = (𝐹 ∘ ((𝐺 ↾ 𝑡) ∘ ^◡(𝐺 ↾ 𝑡)))
82		f1ores 6869	. . . . . . . . . . . . . . 15 ⊢ ((𝐺:(1...(𝑁 + 1))⟶1-1𝐴 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (𝐺 ↾ 𝑡):𝑡⟶1-1onto(𝐺 “ 𝑡))
83	13, 82	sylan 592	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (𝐺 ↾ 𝑡):𝑡⟶1-1onto(𝐺 “ 𝑡))
84		f1ococnv2 6883	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ↾ 𝑡):𝑡⟶1-1onto(𝐺 “ 𝑡) → ((𝐺 ↾ 𝑡) ∘ ^◡(𝐺 ↾ 𝑡)) = ( I ↾ (𝐺 “ 𝑡)))
85	83, 84	syl 18	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → ((𝐺 ↾ 𝑡) ∘ ^◡(𝐺 ↾ 𝑡)) = ( I ↾ (𝐺 “ 𝑡)))
86	85	coeq2d 5851	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (𝐹 ∘ ((𝐺 ↾ 𝑡) ∘ ^◡(𝐺 ↾ 𝑡))) = (𝐹 ∘ ( I ↾ (𝐺 “ 𝑡))))
87		coires1 6288	. . . . . . . . . . . 12 ⊢ (𝐹 ∘ ( I ↾ (𝐺 “ 𝑡))) = (𝐹 ↾ (𝐺 “ 𝑡))
88	86, 87	eqtrdi 2814	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (𝐹 ∘ ((𝐺 ↾ 𝑡) ∘ ^◡(𝐺 ↾ 𝑡))) = (𝐹 ↾ (𝐺 “ 𝑡)))
89	81, 88	eqtrid 2810	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (((𝐹 ∘ 𝐺) ↾ 𝑡) ∘ ^◡(𝐺 ↾ 𝑡)) = (𝐹 ↾ (𝐺 “ 𝑡)))
90		isoeq1 7356	. . . . . . . . . 10 ⊢ ((((𝐹 ∘ 𝐺) ↾ 𝑡) ∘ ^◡(𝐺 ↾ 𝑡)) = (𝐹 ↾ (𝐺 “ 𝑡)) → ((((𝐹 ∘ 𝐺) ↾ 𝑡) ∘ ^◡(𝐺 ↾ 𝑡)) Isom < , < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡))))
91	89, 90	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → ((((𝐹 ∘ 𝐺) ↾ 𝑡) ∘ ^◡(𝐺 ↾ 𝑡)) Isom < , < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡))))
92		imaco 6271	. . . . . . . . . 10 ⊢ ((𝐹 ∘ 𝐺) “ 𝑡) = (𝐹 “ (𝐺 “ 𝑡))
93		isoeq5 7360	. . . . . . . . . 10 ⊢ (((𝐹 ∘ 𝐺) “ 𝑡) = (𝐹 “ (𝐺 “ 𝑡)) → ((𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡)))))
94	92, 93	ax-mp 5	. . . . . . . . 9 ⊢ ((𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡))))
95	91, 94	bitrdi 290	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → ((((𝐹 ∘ 𝐺) ↾ 𝑡) ∘ ^◡(𝐺 ↾ 𝑡)) Isom < , < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡)))))
96	77, 95	sylibd 242	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡)) → (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡)))))
97	47, 96	anim12d 621	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → ((𝑅 ≤ (♯‘𝑡) ∧ ((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡))) → (𝑅 ≤ (♯‘(𝐺 “ 𝑡)) ∧ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡))))))
98	45	breq2d 5120	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (𝑆 ≤ (♯‘(𝐺 “ 𝑡)) ↔ 𝑆 ≤ (♯‘𝑡)))
99	98	biimprd 251	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (𝑆 ≤ (♯‘𝑡) → 𝑆 ≤ (♯‘(𝐺 “ 𝑡))))
100		isotr 7375	. . . . . . . . . 10 ⊢ ((^◡(𝐺 ↾ 𝑡) Isom < , < ((𝐺 “ 𝑡), 𝑡) ∧ ((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , ^◡ < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡))) → (((𝐹 ∘ 𝐺) ↾ 𝑡) ∘ ^◡(𝐺 ↾ 𝑡)) Isom < , ^◡ < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡)))
101	100	ex 418	. . . . . . . . 9 ⊢ (^◡(𝐺 ↾ 𝑡) Isom < , < ((𝐺 “ 𝑡), 𝑡) → (((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , ^◡ < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡)) → (((𝐹 ∘ 𝐺) ↾ 𝑡) ∘ ^◡(𝐺 ↾ 𝑡)) Isom < , ^◡ < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡))))
102	74, 101	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , ^◡ < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡)) → (((𝐹 ∘ 𝐺) ↾ 𝑡) ∘ ^◡(𝐺 ↾ 𝑡)) Isom < , ^◡ < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡))))
103		isoeq1 7356	. . . . . . . . . 10 ⊢ ((((𝐹 ∘ 𝐺) ↾ 𝑡) ∘ ^◡(𝐺 ↾ 𝑡)) = (𝐹 ↾ (𝐺 “ 𝑡)) → ((((𝐹 ∘ 𝐺) ↾ 𝑡) ∘ ^◡(𝐺 ↾ 𝑡)) Isom < , ^◡ < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ^◡ < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡))))
104	89, 103	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → ((((𝐹 ∘ 𝐺) ↾ 𝑡) ∘ ^◡(𝐺 ↾ 𝑡)) Isom < , ^◡ < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ^◡ < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡))))
105		isoeq5 7360	. . . . . . . . . 10 ⊢ (((𝐹 ∘ 𝐺) “ 𝑡) = (𝐹 “ (𝐺 “ 𝑡)) → ((𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ^◡ < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ^◡ < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡)))))
106	92, 105	ax-mp 5	. . . . . . . . 9 ⊢ ((𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ^◡ < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ^◡ < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡))))
107	104, 106	bitrdi 290	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → ((((𝐹 ∘ 𝐺) ↾ 𝑡) ∘ ^◡(𝐺 ↾ 𝑡)) Isom < , ^◡ < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ^◡ < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡)))))
108	102, 107	sylibd 242	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , ^◡ < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡)) → (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ^◡ < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡)))))
109	99, 108	anim12d 621	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → ((𝑆 ≤ (♯‘𝑡) ∧ ((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , ^◡ < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡))) → (𝑆 ≤ (♯‘(𝐺 “ 𝑡)) ∧ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ^◡ < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡))))))
110	97, 109	orim12d 979	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (((𝑅 ≤ (♯‘𝑡) ∧ ((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡))) ∨ (𝑆 ≤ (♯‘𝑡) ∧ ((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , ^◡ < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡)))) → ((𝑅 ≤ (♯‘(𝐺 “ 𝑡)) ∧ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡)))) ∨ (𝑆 ≤ (♯‘(𝐺 “ 𝑡)) ∧ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ^◡ < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡)))))))
111		fveq2 6916	. . . . . . . . 9 ⊢ (𝑠 = (𝐺 “ 𝑡) → (♯‘𝑠) = (♯‘(𝐺 “ 𝑡)))
112	111	breq2d 5120	. . . . . . . 8 ⊢ (𝑠 = (𝐺 “ 𝑡) → (𝑅 ≤ (♯‘𝑠) ↔ 𝑅 ≤ (♯‘(𝐺 “ 𝑡))))
113		reseq2 5981	. . . . . . . . . 10 ⊢ (𝑠 = (𝐺 “ 𝑡) → (𝐹 ↾ 𝑠) = (𝐹 ↾ (𝐺 “ 𝑡)))
114		isoeq1 7356	. . . . . . . . . 10 ⊢ ((𝐹 ↾ 𝑠) = (𝐹 ↾ (𝐺 “ 𝑡)) → ((𝐹 ↾ 𝑠) Isom < , < (𝑠, (𝐹 “ 𝑠)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < (𝑠, (𝐹 “ 𝑠))))
115	113, 114	syl 18	. . . . . . . . 9 ⊢ (𝑠 = (𝐺 “ 𝑡) → ((𝐹 ↾ 𝑠) Isom < , < (𝑠, (𝐹 “ 𝑠)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < (𝑠, (𝐹 “ 𝑠))))
116		isoeq4 7359	. . . . . . . . 9 ⊢ (𝑠 = (𝐺 “ 𝑡) → ((𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < (𝑠, (𝐹 “ 𝑠)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < ((𝐺 “ 𝑡), (𝐹 “ 𝑠))))
117		imaeq2 6066	. . . . . . . . . 10 ⊢ (𝑠 = (𝐺 “ 𝑡) → (𝐹 “ 𝑠) = (𝐹 “ (𝐺 “ 𝑡)))
118		isoeq5 7360	. . . . . . . . . 10 ⊢ ((𝐹 “ 𝑠) = (𝐹 “ (𝐺 “ 𝑡)) → ((𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < ((𝐺 “ 𝑡), (𝐹 “ 𝑠)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡)))))
119	117, 118	syl 18	. . . . . . . . 9 ⊢ (𝑠 = (𝐺 “ 𝑡) → ((𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < ((𝐺 “ 𝑡), (𝐹 “ 𝑠)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡)))))
120	115, 116, 119	3bitrd 308	. . . . . . . 8 ⊢ (𝑠 = (𝐺 “ 𝑡) → ((𝐹 ↾ 𝑠) Isom < , < (𝑠, (𝐹 “ 𝑠)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡)))))
121	112, 120	anbi12d 644	. . . . . . 7 ⊢ (𝑠 = (𝐺 “ 𝑡) → ((𝑅 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , < (𝑠, (𝐹 “ 𝑠))) ↔ (𝑅 ≤ (♯‘(𝐺 “ 𝑡)) ∧ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡))))))
122	111	breq2d 5120	. . . . . . . 8 ⊢ (𝑠 = (𝐺 “ 𝑡) → (𝑆 ≤ (♯‘𝑠) ↔ 𝑆 ≤ (♯‘(𝐺 “ 𝑡))))
123		isoeq1 7356	. . . . . . . . . 10 ⊢ ((𝐹 ↾ 𝑠) = (𝐹 ↾ (𝐺 “ 𝑡)) → ((𝐹 ↾ 𝑠) Isom < , ^◡ < (𝑠, (𝐹 “ 𝑠)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ^◡ < (𝑠, (𝐹 “ 𝑠))))
124	113, 123	syl 18	. . . . . . . . 9 ⊢ (𝑠 = (𝐺 “ 𝑡) → ((𝐹 ↾ 𝑠) Isom < , ^◡ < (𝑠, (𝐹 “ 𝑠)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ^◡ < (𝑠, (𝐹 “ 𝑠))))
125		isoeq4 7359	. . . . . . . . 9 ⊢ (𝑠 = (𝐺 “ 𝑡) → ((𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ^◡ < (𝑠, (𝐹 “ 𝑠)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ^◡ < ((𝐺 “ 𝑡), (𝐹 “ 𝑠))))
126		isoeq5 7360	. . . . . . . . . 10 ⊢ ((𝐹 “ 𝑠) = (𝐹 “ (𝐺 “ 𝑡)) → ((𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ^◡ < ((𝐺 “ 𝑡), (𝐹 “ 𝑠)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ^◡ < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡)))))
127	117, 126	syl 18	. . . . . . . . 9 ⊢ (𝑠 = (𝐺 “ 𝑡) → ((𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ^◡ < ((𝐺 “ 𝑡), (𝐹 “ 𝑠)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ^◡ < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡)))))
128	124, 125, 127	3bitrd 308	. . . . . . . 8 ⊢ (𝑠 = (𝐺 “ 𝑡) → ((𝐹 ↾ 𝑠) Isom < , ^◡ < (𝑠, (𝐹 “ 𝑠)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ^◡ < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡)))))
129	122, 128	anbi12d 644	. . . . . . 7 ⊢ (𝑠 = (𝐺 “ 𝑡) → ((𝑆 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , ^◡ < (𝑠, (𝐹 “ 𝑠))) ↔ (𝑆 ≤ (♯‘(𝐺 “ 𝑡)) ∧ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ^◡ < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡))))))
130	121, 129	orbi12d 932	. . . . . 6 ⊢ (𝑠 = (𝐺 “ 𝑡) → (((𝑅 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , < (𝑠, (𝐹 “ 𝑠))) ∨ (𝑆 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , ^◡ < (𝑠, (𝐹 “ 𝑠)))) ↔ ((𝑅 ≤ (♯‘(𝐺 “ 𝑡)) ∧ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡)))) ∨ (𝑆 ≤ (♯‘(𝐺 “ 𝑡)) ∧ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ^◡ < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡)))))))
131	130	rspcev 3580	. . . . 5 ⊢ (((𝐺 “ 𝑡) ∈ 𝒫 𝐴 ∧ ((𝑅 ≤ (♯‘(𝐺 “ 𝑡)) ∧ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡)))) ∨ (𝑆 ≤ (♯‘(𝐺 “ 𝑡)) ∧ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ^◡ < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡)))))) → ∃𝑠 ∈ 𝒫 𝐴((𝑅 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , < (𝑠, (𝐹 “ 𝑠))) ∨ (𝑆 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , ^◡ < (𝑠, (𝐹 “ 𝑠)))))
132	33, 110, 131	syl6an 697	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (((𝑅 ≤ (♯‘𝑡) ∧ ((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡))) ∨ (𝑆 ≤ (♯‘𝑡) ∧ ((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , ^◡ < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡)))) → ∃𝑠 ∈ 𝒫 𝐴((𝑅 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , < (𝑠, (𝐹 “ 𝑠))) ∨ (𝑆 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , ^◡ < (𝑠, (𝐹 “ 𝑠))))))
133	20, 132	sylan2b 606	. . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝒫 (1...(𝑁 + 1))) → (((𝑅 ≤ (♯‘𝑡) ∧ ((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡))) ∨ (𝑆 ≤ (♯‘𝑡) ∧ ((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , ^◡ < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡)))) → ∃𝑠 ∈ 𝒫 𝐴((𝑅 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , < (𝑠, (𝐹 “ 𝑠))) ∨ (𝑆 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , ^◡ < (𝑠, (𝐹 “ 𝑠))))))
134	133	rexlimdva 3167	. 2 ⊢ (𝜑 → (∃𝑡 ∈ 𝒫 (1...(𝑁 + 1))((𝑅 ≤ (♯‘𝑡) ∧ ((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡))) ∨ (𝑆 ≤ (♯‘𝑡) ∧ ((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , ^◡ < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡)))) → ∃𝑠 ∈ 𝒫 𝐴((𝑅 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , < (𝑠, (𝐹 “ 𝑠))) ∨ (𝑆 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , ^◡ < (𝑠, (𝐹 “ 𝑠))))))
135	19, 134	mpd 16	1 ⊢ (𝜑 → ∃𝑠 ∈ 𝒫 𝐴((𝑅 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , < (𝑠, (𝐹 “ 𝑠))) ∨ (𝑆 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , ^◡ < (𝑠, (𝐹 “ 𝑠)))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 = wceq 1570 ∈ wcel 2147 ∀wral 3080 ∃wrex 3090 Vcvv 3454 ⊆ wss 3900 𝒫 cpw 4560 class class class wbr 5108 I cid 5552 Or wor 5565 ^◡ccnv 5657 ran crn 5659 ↾ cres 5660 “ cima 5661 ∘ ccom 5662 ⟶wf 6562 ⟶1-1wf1 6563 ⟶1-1ontowf1o 6565 ‘cfv 6566 Isom wiso 6567 (class class class)co 7451 ≈ cen 9008 Fincfn 9011 ℝcr 11238 1c1 11240 + caddc 11242 · cmul 11244 < clt 11382 ≤ cle 11383 − cmin 11580 ℕcn 12375 ℕ₀cn0 12646 ...cfz 13679 ♯chash 14515

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5238 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-cnex 11295 ax-resscn 11296 ax-1cn 11297 ax-icn 11298 ax-addcl 11299 ax-addrcl 11300 ax-mulcl 11301 ax-mulrcl 11302 ax-mulcom 11303 ax-addass 11304 ax-mulass 11305 ax-distr 11306 ax-i2m1 11307 ax-1ne0 11308 ax-1rid 11309 ax-rnegex 11310 ax-rrecex 11311 ax-cnre 11312 ax-pre-lttri 11313 ax-pre-lttrn 11314 ax-pre-ltadd 11315 ax-pre-mulgt0 11316 ax-pre-sup 11317

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4841 df-uni 4873 df-int 4913 df-iun 4958 df-br 5109 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-pred 6330 df-ord 6391 df-on 6392 df-lim 6393 df-suc 6394 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-isom 6575 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-om 7912 df-1st 8035 df-2nd 8036 df-frecs 8337 df-wrecs 8368 df-recs 8417 df-rdg 8456 df-1o 8514 df-oadd 8518 df-er 8755 df-en 9012 df-dom 9013 df-sdom 9014 df-fin 9015 df-sup 9471 df-dju 10019 df-card 10057 df-pnf 11384 df-mnf 11385 df-xr 11386 df-ltxr 11387 df-le 11388 df-sub 11582 df-neg 11583 df-nn 12376 df-n0 12647 df-xnn0 12720 df-z 12734 df-uz 13006 df-fz 13680 df-hash 14516

This theorem is used by: erdsze2 36178

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